For the capacitor the choice is relatively easy. Good ceramic capacitors (e.g. TDK C0G) get lower dielectric absorbtion than most film types (maybe except PTFE) and they are small and affordable in the usual size needed (e.g. < 10 nF) for a multi-slope ADC.

For the integrator swing, more swing allows for a smaller capacitor and less voltage noise for the final / initial charge. If this is really relevant depends - with 1 PLC and slower it often is not that imporant.

Classical the swing was some +-10 V max, so that amplifiers with a +-15 V supply were sufficient. I would not consider the swing a very important parameter. For tests it is always helpful to have some headroom for variations. A small integrator capacitor also makes parasitic capacitance more relevant.

For the actual integrator the logic choice today is a combination of 2 OP-amps, to keep the input voltage really close to zero. This is one precision OP-amp and a fast amplifier for the output. I see no more need for the 3 amplifier version as in the 3458, as there are now fast enough precision amplifiers (e.g. OPA140).

A bigger question is what type of ADC to build with different option for the modulation and the choice of run-down phase or charge measurement on the fly for contineous integration.

This choice also effects the choice for the reference switches: rel. slow modulation can use current type switching at the integrator side (e.g. like 34401), while fast modulation works better with voltage side switching (like 3446x).

For the modulation it is mainly the choice between

1) classic 2 case modulaton like in 34401,3458,K200x, LD120 or

2) variable PWM like FB (e.g. Solarton, KS3446x).

Usually there is no real need to model the switches in details. The point is keeping the number of switching events constant, so that charge injection only gives an offset that is easy to subtract.

The noise part is relatively easy, though there can be quite some noise sources and in an optimized design several source tend to be comparable.

The stability is mainly a thing of the parts used, especially the resistors.

The linearity part is difficult, as it is about small non ideal effects, beyond the normal linear modelling and is easy to miss parts.